1. T. E. Dyhoum1, D. Lesnic 1 and R. G. Aykroyd 2
The Solution of Direct and Inverse Complete-Electrode Inverse Model of EIT
T. E. Dyhoum1, D. Lesnic 1 and R. G. Aykroyd 2
1Department of Applied Mathematics, 2Department of Statistics, Leeds University, UK
Purpose
Let denote the boundary collocation points uniformly distributed across the outer boundary .When we apply the boundary condition (2) for the electrodes at the collocation points, this yields
where This yields equations. Furthermore, applying the zero flux boundary condition (3) on the gaps between electrodes, result in
where This yields another equations. Also, we apply (4) which gives more equations
where are collocation points uniformly distributed over Lastly, imposing the condition (5) and using (9), result in one more equation
Equations (7)-(10) can be rewritten as a linear system of algebraic equations
The least-squares method is used to solve the system of equations (10).
Case ii
Figures 4. A star-shaped model with full posterior distribution with chosen prior parameter using the current pattern The estimated radius is
The big Question
If we want to detect the size, shape, position of an organ inside the body, how can we do so?
Introduction
Case iii
In various applications of electrical impedance tomography (EIT) such as medical imaging or geophysics, the purpose is to evaluate the impedance distribution of an object. This means to reconstruct the conductivity of the whole domain using some electrical measurements which are taken on the surface of the object (or body). This task can be achieved by setting a finite number of electrodes which are attached to the outside boundary of that object. Then, the currents are injected through these electrodes. After that the voltages can be calculated using Maxwell's equations which are equivalent to a solution of Laplace's equation, as described in [2]. Herein, the method of fundamental solutions (MFS) is used to numerically calculate the boundary voltages. However, using these voltages as a data set to estimate the interior conductivity distribution results in an ill-posed inverse problem, which needs to be regularized to obtain a stable and reliable solution. Here we use the Bayesian process..
Figures 5. Circle model with full posterior distribution with chosen prior parameter using the single current pattern (12) that results in four voltages. The estimated radius is
Figures 1. Comparison between the MFS solutions for the boundary potential
(left) and the current (right).
Using two current patterns that results in eight voltages. The estimated radius is
Using three current pattern that results in twelve voltages. The estimated radius is
Mathematical Formulation
Statistical Approach
We consider solving the Laplace's equation in a two-dimensional doubly-connected bounded domain of uniform unit conductivity, namely,
subject to boundary conditions which makes the problem the so-called `complete-electrode model` (CEM), [3]. In this model, on the boundary there are attached electrodes, for
. On these electrodes we have the Robin boundary condition,
where is the outward unit normal to the boundary, is the length of the electrode. The current is assumed to vanish between the electrodes on the boundary part, so that
Assuming that is a perfectly conductive rigid inclusion having infinite (very large) conductivity we have the homogeneous Dirichlet boundary condition
In order to obtain a unique solution we also need that,
Equations (1)-(5) represent the direct problem of EIT in the domain containing a rigid inclusion
Conclusions
The purpose behind using the statistical techniques is to reformulate the inverse problem in the form of statistical investigation (inference) seeking for information. The quantities which cannot be observed and depend on each other need to be models unlike those which easily can be observed. So, the goal here is extracting as much as information about these quantities from the boundary data. The uncertainty of the values of any random variables which is needed to be modelled is considered as a probability distribution. By using statistical inversion methods, the solution of the inverse problem is called the posterior distribution of the examined parameter contingent on the measurement. This posterior distribution is obtained from a likelihood combined with a prior distribution using Bayes theorem and Markov chain Monte Carlo method (MCMC) allows to deeply analyze the posterior distribution, not only calculate the mean and the standard deviation, but also estimate the unknown distribution from the sample histogram, for more details see [2].
Figures 2. Equipotential lines of the interior solutions for the current pattern (11) (left) and (12) (right).
This poster has successfully detected the shape, location and size of the rigid inclusion (object), the accuracy is clear form Example 1 and 2. We solved inverse EIT problem based on the MFS direct solver which provides accurate boundary solutions and voltages in combination with the MCMC estimation.
Example 2. Detect the shape, position and the size of the rigid inclusion by solving the inverse EIT problem in the following cases:
Fitting a circular or a star-shaped obstacle using boundary potential and current flux when
Fitting a star-shaped obstacle form circular data using boundary potential and current flux when
Fitting a circular object using voltages with different injected current patterns when
Solution:
References
[1] R. G. Aykroyd, D. Lesnic and A. Karageorghis, (2013) A Bayesian approach for shape estimation of objects, 4th Inverse Problems, Design and Opimization Symposium (IPDO 2013), (eds. O. Fudym, J.-L. Battaglia, G.S. Dulikravich, H.R.B. Orlande and M.J. Colaco), Ecole des Mines d'Albi-Carmaux, Albi, France, Paper (10 pages).
[2] J. P. Kaipio, V. Kolehmainen, E. Somersalo and M. Vauhkonen, Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography, Inverse Problems, 16, (2000), pp
[3] E. Somersalo, M. Cheney and D. Isaacson, Existence and uniqueness for electrode models for electric current computed tomography, SIAM Journal on Applied Mathematics, 52, (1992), pp
Case i
Results and Discussion
Figures 3. Circle model with full posterior distribution with chosen prior parameter:
errors estimation (left), object boundary histogram (middle) and object boundary reliable interval (right). The estimated radius is
We will detect the shape and size of a rigid inclusion based on inverting numerically simulated data .
Example 1. Solve the direct complete electrode model of EIT problem using the MFS for
and with following current patterns
and
Solution:
MFS for the Direct Problem
Star-shaped model with full posterior distribution with three different prior parameters to obtain the best estimation of rigid inclusion. The errors are more assembled in the top-left of the reconstructed object.
The MFS seeks a solution of Laplace's equation (1) as a linear combination of fundamental solutions , in the form:
where and are the internal and external sources which are located inside
and outside respectively. The are the MFS unknown coefficients to be determined by imposing the boundary conditions (1)- (5).
Acknowledgement
T. E. Dyhoum would like to thank the Libyan Ministry of Higher Education and Libyan Embassy for their financial support in this research.